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Dispersion in Glass and Water: Critical Angle
Dispersion in Glass and Water: Critical Angle
Dispersion in Glass and Water: Critical Angle 3
Theory
Generally when white light passes through a material it disperses, or spreads out into a spectrum of component colors, whenever the index of refraction of the material depends on wavelength. A dispersion curve shows the dependence of the index of refraction on wavelength. In this experiment you will determine the dispersion curves of light in glass and water by measuring the critical angle. This method forms the basis for various direct-reading commercial refractometers, such as the Abbe and Pulfrich types.
Figure1 shows the arrangement of the apparatus. Because the collimator is not used, it is turned out of the way. A monochromatic light source diffusely illuminates the glass. The light striking one face of the prism enters the glass and refracts, leaving the prism through another face, where it is observed through the telescope. The angle between these two faces is the angle, A, in the figure.
Recall that a light ray travelling in glass will only refract out of the glass if the incident angle is less than the critical angle. A light ray incident at an angle larger than the critical angle will reflect back into the glass. This process of total internal reflection will be seen in the Fiber Optics experiment. A ray that hits the surface at the precisely the critical angle would refract parallel to the surface. Conversely, the prism will refract all light rays incident on it, but the refracted ray will include all angles up to the critical angle, but not exceeding it. A ray that hits the surface precisely parallel to the surface would refract at the critical angle.
Figure2 shows a more detailed diagram of the light rays. Ray#1 (dashed) represents any of the rays that strike the upper prism face (b) at an angle of incidence less than 90° (measured relative to the normal direction) and, when refracted, contribute to a broad band seen in the telescope. The edge of the band is determined by the limiting ray, ray#2 (solid), that strikes the upper prism face at a glancing angle and refracts at the critical angle rc. These rays strike the second prism face (c) at angle r', emerge at angle i', and form a beam that the telescope images as the sharp edge of the broad band of light.
Knowing the angles involved will allow you to calculate the index of refraction. From the geometry of Figure2,
(1)
Also, from Snell's law,
(2)
and
(3)
where Nl is the index of refraction of the liquid and Ng is that of the glass. Combining Equations1, 2, and 3 by elimination of rc and r' yields
(4)
In Equation4, angle i' could be negative, indicating that the ray emerges on the opposite side of the
normal to prism face c. In this case, sin i' would be negative.
If there is no liquid on the upper prism face (b), then Nl is actually the index of refraction of air (assumed to be 1) and Equation4 may be solved for Ng:
(5)
After you use Equation5 to calculate the index of refraction of the prism, you will use Equation4 to calculate the index of refraction of the liquid.
Procedure
The spectrometer should already be aligned and the telescope should already be properly focussed. Otherwise, seek help from your instructor.
Measuring Prism Angle A
Arrange the prism and the collimator as shown in Figure3. Use the sodium light source. Swing the telescope to the position, T, where you see the reflection of the light source off of the face of the prism. Align the crosshairs with the reflection and record the angle reading from both verniers. Now swing the telescope to position T'' and repeat the measurement. Then, the angle between T and T'' is equal to 2A. Calculate A for both sets of vernier readings and compute the average; the result should be about 60°.
Index of Refraction of Glass
Arrange the apparatus as shown in Figure1, but do not put the water and the ground glass plate on the face of the prism. Use the sodium light source. Place the ground glass screen between the source and the prism to diffuse the light.
A piece of black paper held to the prism with a small piece of masking tape serves as a light shield. Position the shield so that it blocks all rays through
the unused face but does not block the rays that hit the upper face of the prism at grazing incidence. Also, align the prism so that as much light as possible hits the face at grazing incidence; use the shadow of the edge of the prism to guide you.
Figure 3. Measuring Prism Angle A
Look through the telescope and locate the sharp boundary corresponding to the critical angle i'. The angular position of the boundary is independent of the position of the light source with respect to the prism, within wide limits. The sharpness of the boundary, however, depends on the illumination; take care to make the boundary as sharp as possible by adjusting the position of the light source.
Now, swing the telescope so that it is perpendicular to the prism face c. Plug the light cord for the telescope eyepiece into the power strip. Some of the light will illuminate the crosshairs, while some of it will reflect off of a mirror (see Figure4) towards the prism face. This reflected light will in turn reflect off of the prism, return through the telescope, and illuminate the crosshairs from behind, creating a dark image of the crosshairs. This dark image will be aligned with the crosshairs only if the telescope is pointed normal to the prism face. Position the telescope so that the image and the crosshairs coincide, and read the angle on both verniers. If the image is displaced vertically from the crosshairs, you need to adjust the tilt of the spectrometer platform. Unplug the light cord for the eyepiece when you have completed this measurement.
Swing the telescope to align the critical angle boundary with the crosshairs and record the readings for both verniers. The difference between the readings at the critical angle and the readings at the normal position is then the angle i' in Equation5.
Substitute the mercury light source for the sodium light source. When using the mercury light source, do not look into the source; use the aluminum foil to shield the room from the source and keep the dangerous ultraviolet light out of your eyes. For measurements with the mercury source, Place the filters between the mercury source and the glass plate.
The mercury light source, unlike the sodium light source, yields several different bright emission lines. Hence, you will see a series of color bands with different critical angles. To aid in identifying the critical angle boundary for a particular wavelength, insert the holder with one of the five filters between the mercury source and the ground glass plate. Do not remove the ground glass plate.
Find the cutoff angle for each wavelength (see Table 1).
Table 1Filter / Color / Wavelength (nm)
2 / Red / 623.4
4 / Yellow / 578.0
7 / Green / 546.1
9 / Blue / 491.6
12 / Violet / 435.8
(In some cases, you may still see two bright bands.) Calculate the value of the index of refraction of glass, Ng, for all six wavelengths you have used.
Plot the dispersion curve for glass; that is, plot the index of refraction of glass as a function of wavelength.
Index of Refraction of Water
Now put a few drops of distilled water on a ground glass plate and carefully put the plate on the side of the prism, as shown in Figure1, without moving the prism. The edge of the glass plate should be flush with the unused face of the prism. Again, position the sodium light source so that as much of the light as possible hits the prism at grazing incidence without reflecting off of the glass plate, and determine the critical angle. Use Equation4 to calculate the index of refraction of water, Nl. Find the index of refraction of water for just the sodium light source.
Compare the value of Nl found for the sodium light source with the accepted value of 1.33299 and calculate the percent error.
Analysis
From the dispersion curve for glass, compute the dispersive power of glass, by determining the quantity
, (6)
where NF, NC, and ND are the indices of refraction at the wavelengths of the Fraunhofer F, C, and D lines: λF = 486.1 nm, λD = 589.0 nm, λC = 656.3 nm.
Discuss the sources of error in this experiment and evaluate their significance.
Justify equation 1, 2, and 3, and derive Eqs 4 and 5 from them.
Dispersion in Glass and Water: Critical Angle 3